# Properties

 Label 600.2.r.b Level $600$ Weight $2$ Character orbit 600.r Analytic conductor $4.791$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.r (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( 1 - 2 \zeta_{8} + \zeta_{8}^{2} ) q^{7} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( 1 - 2 \zeta_{8} + \zeta_{8}^{2} ) q^{7} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{11} + ( -1 + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{13} + ( 1 - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{17} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} + ( 1 + 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{21} + ( 3 - 2 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{23} + ( 5 + \zeta_{8} + \zeta_{8}^{3} ) q^{27} + ( 2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{31} + ( 4 + 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{33} + ( 3 - 4 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{37} + ( 5 - 2 \zeta_{8} + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{39} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{41} + ( -1 + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( 5 - 5 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{47} + ( -4 \zeta_{8} - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{49} + ( 3 + 2 \zeta_{8} + 5 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{51} + ( -3 - 3 \zeta_{8}^{2} ) q^{53} + ( -4 - 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{57} + 4 q^{59} + ( 6 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{61} + ( -5 + 2 \zeta_{8} + 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + ( -3 + 10 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{67} + ( -1 + 2 \zeta_{8} - 5 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{69} + ( -6 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{71} + ( -1 + \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{73} + ( -2 + 2 \zeta_{8}^{2} ) q^{77} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{79} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + ( -1 - 6 \zeta_{8} - \zeta_{8}^{2} ) q^{83} + ( -2 - 2 \zeta_{8} + 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{87} + ( -10 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{89} + ( -10 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{91} + ( 4 \zeta_{8} - 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{93} + ( -1 - \zeta_{8}^{2} ) q^{97} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} + 4 q^{7} - 4 q^{9} - 4 q^{13} + 4 q^{17} + 4 q^{21} + 12 q^{23} + 20 q^{27} + 8 q^{29} + 16 q^{33} + 12 q^{37} + 20 q^{39} - 4 q^{43} + 20 q^{47} + 12 q^{51} - 12 q^{53} - 16 q^{57} + 16 q^{59} + 24 q^{61} - 20 q^{63} - 12 q^{67} - 4 q^{69} - 4 q^{73} - 8 q^{77} - 28 q^{81} - 4 q^{83} - 8 q^{87} - 40 q^{89} - 40 q^{91} - 4 q^{97} - 32 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$\zeta_{8}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 −1.00000 1.41421i 0 0 0 2.41421 + 2.41421i 0 −1.00000 + 2.82843i 0
257.2 0 −1.00000 + 1.41421i 0 0 0 −0.414214 0.414214i 0 −1.00000 2.82843i 0
593.1 0 −1.00000 1.41421i 0 0 0 −0.414214 + 0.414214i 0 −1.00000 + 2.82843i 0
593.2 0 −1.00000 + 1.41421i 0 0 0 2.41421 2.41421i 0 −1.00000 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.r.b 4
3.b odd 2 1 600.2.r.c 4
4.b odd 2 1 1200.2.v.j 4
5.b even 2 1 120.2.r.c yes 4
5.c odd 4 1 120.2.r.b 4
5.c odd 4 1 600.2.r.c 4
12.b even 2 1 1200.2.v.d 4
15.d odd 2 1 120.2.r.b 4
15.e even 4 1 120.2.r.c yes 4
15.e even 4 1 inner 600.2.r.b 4
20.d odd 2 1 240.2.v.a 4
20.e even 4 1 240.2.v.c 4
20.e even 4 1 1200.2.v.d 4
40.e odd 2 1 960.2.v.i 4
40.f even 2 1 960.2.v.a 4
40.i odd 4 1 960.2.v.f 4
40.k even 4 1 960.2.v.g 4
60.h even 2 1 240.2.v.c 4
60.l odd 4 1 240.2.v.a 4
60.l odd 4 1 1200.2.v.j 4
120.i odd 2 1 960.2.v.f 4
120.m even 2 1 960.2.v.g 4
120.q odd 4 1 960.2.v.i 4
120.w even 4 1 960.2.v.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.r.b 4 5.c odd 4 1
120.2.r.b 4 15.d odd 2 1
120.2.r.c yes 4 5.b even 2 1
120.2.r.c yes 4 15.e even 4 1
240.2.v.a 4 20.d odd 2 1
240.2.v.a 4 60.l odd 4 1
240.2.v.c 4 20.e even 4 1
240.2.v.c 4 60.h even 2 1
600.2.r.b 4 1.a even 1 1 trivial
600.2.r.b 4 15.e even 4 1 inner
600.2.r.c 4 3.b odd 2 1
600.2.r.c 4 5.c odd 4 1
960.2.v.a 4 40.f even 2 1
960.2.v.a 4 120.w even 4 1
960.2.v.f 4 40.i odd 4 1
960.2.v.f 4 120.i odd 2 1
960.2.v.g 4 40.k even 4 1
960.2.v.g 4 120.m even 2 1
960.2.v.i 4 40.e odd 2 1
960.2.v.i 4 120.q odd 4 1
1200.2.v.d 4 12.b even 2 1
1200.2.v.d 4 20.e even 4 1
1200.2.v.j 4 4.b odd 2 1
1200.2.v.j 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{4} - 4 T_{7}^{3} + 8 T_{7}^{2} + 8 T_{7} + 4$$ $$T_{17}^{4} - 4 T_{17}^{3} + 8 T_{17}^{2} + 56 T_{17} + 196$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 + 8 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$11$ $$16 + 24 T^{2} + T^{4}$$
$13$ $$196 - 56 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$196 + 56 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$16 + 24 T^{2} + T^{4}$$
$23$ $$196 - 168 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$( -28 - 4 T + T^{2} )^{2}$$
$31$ $$( -32 + T^{2} )^{2}$$
$37$ $$4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$4 - 8 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$2116 - 920 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$53$ $$( 18 + 6 T + T^{2} )^{2}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$( 4 - 12 T + T^{2} )^{2}$$
$67$ $$6724 - 984 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$4624 + 152 T^{2} + T^{4}$$
$73$ $$3844 - 248 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$79$ $$16 + 24 T^{2} + T^{4}$$
$83$ $$1156 - 136 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$( 68 + 20 T + T^{2} )^{2}$$
$97$ $$( 2 + 2 T + T^{2} )^{2}$$